More consistency results in partition calculus

This paper has two principal aims. The first is to supply a proof of Theorem 6 of [ShSt1]:Theorem:If ZFC+ "there are c ⁺ measurable cardinals" is consistent, then so is ZFC+ " א _c +is not a strong limit cardinal and {Mathematical expression}". This is done in sections 1 and 2. See the introduction for a discussion of the evolution of the proof and of some interesting questions which remain open, related to obstacles encountered in obtaining maximum freedom in arranging for any desired cardinal exponentiation in Theorems 4 and 6 of [ShSt1]. The method is quite generally applicable in partition calculus and variants of it have in fact been applied in recent work of the authors, see [ShSt2]. first, a preservation result is proved for the game-theoretic properties of the filters considered in [ShSt1]. Then, it is shown that the existence of a system of such filters yields a canonization-style result. Finally, it is shown that the canonization property gives the positive partition relation. The second aim makes the title of this paper slightly inaccurate (but we suspect this will be pardoned): we supply a (straightforward) proof of a result which shows that Theorem 2 of [ShSt1] in some sense is best possible. This is done in section 3.